Differential Equations, what are they? a few ?'s

I'm having a tough time figuring out exactly what differential equations are. after reading the chapter on it 3 times over, I do not understand how one equation is a solution of the diff.eq. of some other equation. Can somebody explain to this to me? I've got 2 problems that I found that would help me learn what it is if somebody can explain.

first one:
show that y = x - x^-1 is a solution of the differential equation xy' + y = 2x

there's only 1 example in my book and after following its same steps, i get y' = (1/x^2) + 1, how is that a solution of xy' + y = 2x?

second:
1) what can be concluded about a solution of the equation y' = -y^2 just by looking at it?
2) verify all members of family y = 1/(x+C) are solutions of y' =-y^2
3) think of a solution NOT a meber of the family in part 2
4) find a solution of the initial-value problem: y' = -y^2 y(0)=.5

the second one really has me wondering of how to approach y' = -y^2 and especially in 2), how is that function a solution?

I'm having a tough time figuring out exactly what differential equations are. after reading the chapter on it 3 times over, I do not understand how one equation is a solution of the diff.eq. of some other equation. Can somebody explain to this to me? I've got 2 problems that I found that would help me learn what it is if somebody can explain.

first one:
show that y = x - x^-1 is a solution of the differential equation xy' + y = 2x

there's only 1 example in my book and after following its same steps, i get y' = (1/x^2) + 1, how is that a solution of xy' + y = 2x?

second:
1) what can be concluded about a solution of the equation y' = -y^2 just by looking at it?
2) verify all members of family y = 1/(x+C) are solutions of y' =-y^2
3) think of a solution NOT a meber of the family in part 2
4) find a solution of the initial-value problem: y' = -y^2 y(0)=.5

the second one really has me wondering of how to approach y' = -y^2 and especially in 2), how is that function a solution?